Classical dynamics, arrow of time, and genesis of the Heisenberg - - PowerPoint PPT Presentation

Classical dynamics, arrow of time, and genesis of the Heisenberg commutation relations Detlev Buchholz & Klaus Fredenhagen

Mathematics of interacting QFT models University of York, July 1st 2019

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Background

Perturbative AQFT led to a new constructive scheme for quantum physics

Ingredients: classical systems, orbits in configuration space, Lagrangeans

• perations (perturbations of system), labelled by functionals on
• rbits (fixed by potentials, durations in time)

arrow (direction) of time; entering into the microworld by order (succession) of operations Result: dynamical C*-algebra for given Lagrangean; commutation relations etc arise from its intrinsic structure. New look at quantum physics; no a priori "quantization rules" This talk: application of scheme to classical mechanics

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Classical mechanics

Notation: N particles in Rs, equal masses, distinguishable positions x = (x1, . . . , xN) ∈ RsN continuous orbits x : R → RsN, family denoted by C , loops x0 : R → RsN with compact support, form vector space C0 ⊂ C , velocities ˙ x : R → RsN Perturbations: (given by potentials, time dependencies) are described by space F of functionals F : C → R F[x] . =

• dt F(t, x(t))

t, x → F(t, x) = f 0(t) x +

• k

gk(t) Vk(x) where f 0 ∈ C0, gk ∈ D(Rs), Vk continuous, bounded Support of F: union of supports of underlying test functions Shifts: F → F x0, x0 ∈ C0 given by x → F x0[x] . = F[x + x0]

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Classical mechanics

Lagrangeans: t → L(x(t)) . = (1/2) ˙ x(t)2 − V(x(t)) , x ∈ C ; action

• dtL(x(t)); relative action for loops x0 ∈ C0:

δL(x0)[x] =

• dt χ(t)

L(x(t) + x0(t)) − L(x(t))

• with χ ↾ supp x0 = 1 (note: element of F, linear term x appears)

Stationary points of action: Euler-Lagrange equation ¨ x(t) + ∂V(x(t)) = 0 Propagators: “inverses” of K = − d2

dt2

i.e. K∆• = ∆•K = 1 advanced ∆A, retarded ∆R, mean ∆D . = (1/2)(∆A + ∆R) commutator function: ∆ . = ∆R − ∆A, K∆ = ∆K = 0

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Dynamical algebra

Step 1: given a Lagrangean L, construct a dynamical group GL Definition: GL is the free group generated by symbols SL(F), F ∈ F, modulo the relations (i) SL(F) = SL(F x0 + δL(x0)) for all F ∈ F, x0 ∈ C0 (ii) SL(F1)SL(F2) = SL(F1 + F2) whenever F1 has support in the future of F2 Remarks: the elements of GL describe the effects of perturbations on the underlying system without stipulating their concrete action (i) F = 0 implies SL(δL(x0)) = S(0) = 1 (Euler-Lagrange equation) (ii) constant functionals Fh : F → h ∈ R have arbitrary support in time; thus S(F)S(Fh) = S(F + Fh) = S(Fh)S(F) (form central subgroup)

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Dynamical algebra

Step 2: proceed from GL to *-algebra AL sums: cS, c ∈ C, S ∈ GL span AL adjoints: ( cS)∗ . = cS−1 (S unitary operators) products: fixed by distributive law fixing scale: S(Fh) = e ih 1, h ∈ R (amounts to atomic units) norm: algebra has faithful states and thus a (maximal) C*-norm Definition: Given L , the corresponding dynamical algebra AL is the C*-algebra determined by the dynamical group GL. No quantization conditions, functional integrals etc ; only classical concepts used (“common language”, cf. Bohr’s doctrine)

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Derivation of Heisenberg relations

Consider non-interacting Lagrangean t → L0(x(t)) = (1/2) ˙ x(t)2 Simplest (linear) perturbations f 0, x . =

• dt f 0(t) x(t)

x → Ff 0[x] . = f 0, x + (1/2)f 0, ∆Df 0 , f 0 ∈ C0 Definition: W(f 0) . = SL0(Ff 0), f 0 ∈ C0.

Theorem

(1) W(Kx0) = 1, x0 ∈ C0 (2) W(f 0)W(g0) = e−(i/2)f 0,∆g0W(f 0 + g0), f 0, g0 ∈ C0 Interpretation: Weyl operators W(x0) . = e ix0,Q, x0 ∈ C0 (1) generators solutions of Heisenberg eq.: t → Q(t) = Q + t ˙ Q (2) [Qk, ˙ Ql] = iδkl1, [Qk, Ql] = [ ˙ Qk, ˙ Ql] = 0 Physics: operators of position Q and momentum P . = ˙ Q

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Derivation of Heisenberg relations

Proofs: (1) (dynamics) x0 ∈ C0

recall: K = − d2

dt2

x → FKx0[x] = Kx0, x + (1/2)Kx0, ∆DKx0 = ˙ x0, ˙ x + (1/2) ˙ x0, ˙ x0 = δL0(x0)[x] hence SL0(FKx0) = SL0(δL0(x0)) = SL0(0) = 1; similarly SL0(Ff 0+Kx0) = SL0(Ff 0) (2) (Weyl relations) Given f 0, g0, let f 0 + Kx0 be later than g0. Then SL0(Ff 0)SL0(Fg0) = SL0(Ff 0+Kx0)SL0(Fg0) = SL0(Ff 0+Kx0 + Fg0). Linearity of Ff[x] with regard to x implies SL0(Ff 0+Kx0 + Fg0) = SL0(Ff 0+Kx0+g0 + Fhf0+Kx0,g0) = e ihf0+Kx0,g0 SL0(Ff 0+Kx0+g0) = e ihf0+Kx0,g0 SL0(Ff 0+g0) where hf 0+Kx0,g0 = · · · = −(1/2)f 0 + Kx0, ∆g0 = −(1/2)f 0, ∆g0 .

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Derivation of Heisenberg relations

Proofs: (1) (dynamics) x0 ∈ C0

recall: K = − d2

dt2

x → FKx0[x] = Kx0, x + (1/2)Kx0, ∆DKx0 = ˙ x0, ˙ x + (1/2) ˙ x0, ˙ x0 = δL0(x0)[x] hence SL0(FKx0) = SL0(δL0(x0)) = SL0(0) = 1; similarly SL0(Ff 0+Kx0) = SL0(Ff 0) (2) (Weyl relations) Given f 0, g0, let f 0 + Kx0 be later than g0. Then SL0(Ff 0)SL0(Fg0) = SL0(Ff 0+Kx0)SL0(Fg0) = SL0(Ff 0+Kx0 + Fg0). Linearity of Ff[x] with regard to x implies SL0(Ff 0+Kx0 + Fg0) = SL0(Ff 0+Kx0+g0 + Fhf0+Kx0,g0) = e ihf0+Kx0,g0 SL0(Ff 0+Kx0+g0) = e ihf0+Kx0,g0 SL0(Ff 0+g0) where hf 0+Kx0,g0 = · · · = −(1/2)f 0 + Kx0, ∆g0 = −(1/2)f 0, ∆g0 .

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Interacting theories

Change of Lagrangean (potentials V as before) t → L(x(t)) = L0(x(t)) − V(x(t)) temporary perturbation (χ smooth characteristic function) t → Lχ(x(t)) . = L0(x(t)) − χ(t)V(x(t))

• Vχ(t,x(t))

Definition: (cf. relative scattering matrices) SLχ(F) . = SL0(−Vχ)−1SL0(F − Vχ) ∈ AL0, F ∈ F Properties: (elementary computation) SLχ(F x0 + δLχ(x0)) = SLχ(F) (i) SLχ(F1)SLχ(F2) = SLχ(F1+F2) if F1 is later than F2 (ii) Conclusion: defining relations for dynamical algebra ALχ≃ AL0

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Interacting theories

Goal: limit χ → 1 (global dynamics) Note: Let I ⊂ R and χ ↾ I = 1, then δLχ(x0) = δL(x0) if x0 ∈ C0(I) Definition: ALχ(I) algebra generated by SLχ(F), F ∈ F(I). Observation: ALχ(I) ≃ AL(I) and algebras ALχ(I) for different χ are related by inner automorphisms of AL0 Detailed analysis: for increasing intervals In and functions χn there exist injective homomorphisms βn : AL(In) → AL0(In+1) such that γ . = lim

n βn

point-wise in norm on AL =

I AL(I).

Theorem

Let L0, L be Lagrangeans. There exist monomorphisms γ : AL → AL0 such that γ(AL(I)) ⊂ AL0( I) for any I and bounded I ⊃ I.

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Representations

Consider Schrödinger representation of Q, P on HS with dynamics L0 Claim: operators S(F) are represented by time ordered exponentials. Problem: For F ∈ F, determine T(F) . = T e i

R ∞

−∞dt F(Q+tP)

• bounded functionals Fb: Dyson expansion

T(Fb) = 1 +

k ik ∞ −∞dt1 . . .

tk−1

−∞ dtk Fb(Q + t1P1) · · · Fb(Q + tkP1)

• linear (unbounded) L: solution of linear differential equation

T(Lf 0) = e i

R dt f 0(t)(Q+tP) e−(i/2)f 0,∆Df 9 = W(f 0) e−(i/2)f 0,∆Df 9

• combination: T(Fb + Lf 0) .

= T(F −∆Af 0

b

) T(Lf 0) Ansatz based on results of structural analysis; it has all required properties

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Representations

Proof: E.g. “dynamical relation” for bounded functionals Fb T(F x0

b + δL0(x0)) = T(F x0 b

+ FKx0) = T(

• F x0

b

+ Fh +LKx0) = T(F x0−∆AKx0

b

) eih T(LKx0) = T(F x0−∆AKx0

b

• Fb

) T(FKx0)

• 1

= T(Fb) Definition: Representation (πS, HS) of AL0 fixed by putting πS(SL0(F)) . = T(F), F ∈ F. Other algebras AL are represented by (π, HS), where π . = πs ◦ γ

Theorem

(i) The representations (π, HS) of AL are “regular” and irreducible (ii) This holds also true for π ↾ AL(I) for any finite interval I

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Observables, statistics and operations

Task: Using only operations, (i) compute probability that a state has the property described by a projection E and (ii) determine properties

• f the state after the operation.

Definition: Let (π, H) be irreducible representation of AL, let Ω ∈ H, and consider vector state ω( · ) = Ω, π( · ) Ω on AL. The operations S ∈ AL induce maps ω → ωS . = ω ◦ Ad S−1 Transition probability: (“fidelity of operation”) ω · ωs . = |Ω, π(S)Ω|2 = |ω(S)|2

Theorem

Let HN ⊂ H be finite dimensional, let E be infinite projection, and let ε > 0. There exists unitary operator Sε ∈ AL such that for any Ω ∈ HN |ω · ωSε − ω(E)2| < ε , ωSε(1 − E) < ε . Note: no collapse of wave functions (Lüders, von Neumann); suitable

• perations determine “primitive observables” [DB, E. Størmer]

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Conclusions

New look at quantum mechanics, based on classical concepts system: configuration space, orbits, Lagrangean

• perations: perturbations of system (with or without observer)

time: directed; its arrow matters already in microphysics Effect of operations on system described by dynamical group (composition of operations) extension to C*-algebra standard procedure no quantization rules; non-commutativity due to arrow of time Consequences commutation relations, familiar framework recovered representation theory based on time ordered products statistical interpretation can be deduced from operations Approach works also in QFT

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Conclusions

New look at quantum mechanics, based on classical concepts system: configuration space, orbits, Lagrangean

• perations: perturbations of system (with or without observer)

time: directed; its arrow matters already in microphysics Effect of operations on system described by dynamical group (composition of operations) extension to C*-algebra standard procedure no quantization rules; non-commutativity due to arrow of time Consequences commutation relations, familiar framework recovered representation theory based on time ordered products statistical interpretation can be deduced from operations Approach works also in QFT: Klaus Fredenhagen

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